Question

Sum to first I,m,n terms of AP.are P, q, r. Prove that p/l(m-n)+q/m(n-l)+I/n(I-m)=0

Answer 1

Let y be the first term and d be the common difference.

= > Given that sum of 1st l,m,n terms of an AP is p,q,r.

We know that sum of n terms of an AP sn = (n/2)[2a + (n - 1) * d].

The answer to this question is explained in the attachment.

Hope it helps!

Answer 2

Dear Student,

Solution:

Let fist term is a and common difference is d.

As we know that sum of n terms in AP = n÷2 ( 2a +(n-1)d)

Here According to question:

Sum of l terms Sl = l÷2( 2a+(l-1)d) = p

( a+(l-1)d/2) = p/l ......eq1

Sum of m terms Sm= m÷2( 2a+(m-1)d) = q

( a+(m-1)d/2) = q/m ......eq2

Sum of n terms Sl = n÷2( 2a+(n-1)d) = r

( a+(n-1)d/2) = r/n .......eq3

To Prove that p/l(m-n)+q/m(n-l)+I/n(I-m)=0 ........eq4

put the value of p/l ,q/m and r/n from eq 1,2 and 3 in eq 4

( a+(l-1)d/2) (m-n) +( a+(m-1)d/2) (n-l) +( a+(n-1)d/2)( l-m)

( a+ld/2 -d/2)(m-n)+( a+md/2 -d/2)(n-l)+( a+nd/2-d/2)(l-m)

am+mld/2-md/2-an-nld/2+nd/2+an+mnd/2-nd/2-al-mld/2+ld/2+al +nld/2-ld/2-am-mnd/2+md/2

= 0

you can see that all the terms present with -ve and +ve values ,cancels each other.

hope it helps you.